#7623204

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622979
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

(2,4)(5,6)(7,17)(8,14)(9,26)(10,13)(11,29)(12,20)(15,18)(16,27)(19,38)(21,37)(22,49)(23,33)(24,52)(25,53)(28,54)(30,48)(31,45)(32,57)(34,43)(35,42)(36,51)(39,41)(40,47)(44,59)(46,66)(50,67)(55,64)(56,68)(58,63)(60,62)(61,65);(5,7,18,41,11,22,28,48,10,21,43,57,27,47,9,20)(6,12,26,40,16,32,34,37,13,30,54,49,29,39,15,17)(8,19,42,62,25,46,50,64,23,44,63,68,51,65,24,45)(14,31,52,61,36,56,58,59,33,55,67,66,53,60,35,38);(1,2,3,4)(6,13)(12,30)(14,33)(15,34)(16,29)(17,37)(26,54)(31,55)(32,39)(35,58)(36,53)(38,59)(40,49)(52,67)(56,60)(61,66);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,31)(13,33)(15,35)(16,36)(17,38)(18,42)(20,45)(21,44)(22,46)(26,52)(27,51)(28,50)(29,53)(30,55)(32,56)(34,58)(37,59)(39,60)(40,61)(41,62)(43,63)(47,65)(48,64)(49,66)(54,67)(57,68);(8,23)(14,33)(19,44)(24,50)(25,51)(31,55)(35,58)(36,53)(38,59)(42,63)(45,64)(46,65)(52,67)(56,60)(61,66)(62,68)

Polycyclic presentation:

2^9|b^2=fh,c^2=gi,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530400);